Research paperThree-dimensional dynamics and synchronization of two coupled fluid-conveying pipes with intermediate springs
Introduction
The phenomenon of synchronization was firstly observed by the Dutch physicist, Huygens, who, in 1665, stumbled upon the perfect synchronization of two side-by-side pendulums, opening a branch in mathematical science known as coupled-oscillator theory. Based on this theory, much attention has been paid to synchronization and its mechanisms, especially in living bodies, such as pacemaker cells in heart, neural networks that control human rhythms, simultaneous lighting and extinguishing of countless fireflies, and cricket singing in unison. Indeed, synchronization has been found to have very attractive applications and huge market potential value in the fields such as physical chemistry [1], [2], biology [3], mechanics [4], [5], [6], brain science [7], [8], [9], electronics, information science and secure communication [10], [11].
Dynamical synchronization of coupled oscillating systems is a typical branch of synchronizations in mechanics and hence has attracted great interest from many scholars. In 1982, based on the extended Lyapunov matrix method, Fujisaka et al. [12] investigated the stability criterion of synchronous vibration of coupled-oscillator systems. In their work, three coupled configurations were considered and the critical modes and state transitions were discussed. Gauthier et al. [13] analyzed a typical experimental system by the use of a commonly used theory for synchronization of coupled-oscillator systems and found that it could fail to estimate the accurate range of high-quality synchronization of the system. They also proposed a novel and feasible approach to predict the regime of high-quality synchronization. By the use of finite difference method, Barrón et al. [14] investigated the dynamical synchronization of a coupled self-excited system consisting of four elastic beams, and found that the coupled self-excited system was robust to dynamically and synchronously change the fluid–structure interaction properties.
It is known that chaos is an important phenomenon in nonlinear dynamics. Therefore, it is of great academic significance to study the synchronization behaviors of coupled chaotic oscillators. By using experimental and numerical methods, Taherion et al. [15] investigated how noise might influence the lag synchronization of two coupled chaotic oscillators. In their work, they found that when the noise in the coupled system is large, it is difficult to generate lag synchronization but a straightforward transition to complete synchronization would occur. Rosenblum et al. [16] investigated the dynamical synchronization transitions of two coupled self-excited oscillators. They found that two coupled self-excited oscillators may experience several transitions as the coupling strength increases. In addition to the study of lag synchronization, Boccaletti et al. [17] tried to introduce several main synchronization types, and analyzed the synchronization characteristics for different systems, including identical, nonidentical, structurally nonequivalent, chaotic and extended systems. Indeed, dynamical synchronization in coupled chaotic systems has been widely discussed in other literature [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].
For the dynamical system of two or multiple coupled pipes, synchronization phenomenon in tube bundle in heat exchangers [28], [29] and multi-channel micro/nano tubes in MEMS was also observed. Because of coupling effects, the coupled fluid-conveying pipes may show wonderful dynamical synchronization behaviors. Ni et al. [30] investigated the dynamical synchronization of two coupled fluid-conveying pipes connected with linear springs. In their research, the effects of three typical fluid flow conditions and coupling stiffness on the dynamical behavior of the coupled two-pipe system were considered. It was found that spring stiffness has an important effect on the synchronization behaviors of the two-pipe system. Based on a similar physical model, Lü et al. [31] studied the dynamical synchronization and bifurcation behaviors of two fluid-conveying pipes connected by a nonlinear spring. In their work, the influences of spring location, spring stiffness and several important flow parameters on the bifurcation results were investigated. In addition, several synchronization patterns were detected in their work, such as lag, perfect, imperfect, strange and failed patterns. Combined with typical dynamic response results, including time histories, phase-plane plots, phase trajectories and PSD diagrams, these synchronization patterns were discussed in detail.
However, it should be pointed out that the vibrations of the two pipes considered in Refs. [30], [31] were assumed to be planar along the spring direction only. In fact, due to the action of the connecting springs, the stiffness of the two pipes along the spring direction is greater. Thus, the two pipes are more likely to vibrate perpendicular to the spring direction. With this consideration, it is necessary to visit the nonplanar vibration problem of the coupled two-pipe system. Moreover, to the best of the authors’ knowledge, the literature on this topic is very limited. Motived by this, the three-dimensional dynamics and synchronization behaviors of the dynamical system of two coupled fluid-conveying pipes connected by linear springs are explored in the present work, with three important issues being focused on: (i) Compared with a single fluid-conveying pipe, what is the difference between the coupled frequencies and modal shapes of the two coupled pipes and a single pipe? (ii) Under the action of the connecting springs, what is the difference between the dynamic responses of the two pipes along the spring direction and perpendicular to the spring direction? (iii) What synchronization and bifurcation behaviors can the two nonlinearly coupled pipes exhibit?
The remainder of this paper is organized as follows. In Section 2, a mathematical model for motions of the coupled two-pipe system is established, and then some key system parameters and initial conditions are given. In Section 3, the dynamic response of a single simply-supported pipe conveying steady fluid predicted in this study is compared with previous results, and good agreement is observed. The convergence analysis for the truncated mode number of Galerkin’s method is then conducted to determine a suitable truncated mode number. In Section 4, modal analysis for the coupled two-pipe system is carried out. By utilizing the linearized form of the governing equations, the natural frequency of the coupled pipe system is analyzed and the modal shapes of the lowest six modes are plotted. At the same time, based on the same physical model and system parameters, the model analysis is also conducted in the FEA (Finite Element Analysis) tool (ANSYS 2019 R3). The present theoretical results are compared with those calculated in the FEA software, and a good agreement is obtained. In Section 5, the nonlinear dynamics and dynamical synchronization of the two coupled fluid-conveying pipes are extensively studied. Finally, some important conclusions are drawn in Section 6.
Section snippets
Model description
The dynamical system of two fluid-conveying pipes under consideration is illustrated in Fig. 1. It consists of two identical pipes, denoted as pipe 1 and pipe 2, connected with three identical springs. The two pipes are of outer/inner diameter , length L, flexural stiffness EI, and conveying steady or pulsatile fluid of velocity and , respectively. The two pipes are simply supported at both ends and their initial centerlines are straight.
As for this two-pipe system, the following
Validation of a single pipe without intermediate spring
In this subsection, the validation of a single fluid-conveying pipe without intermediate springs will be checked by comparing the present result with that of Modarres-Sadeghi et al. [35]. In the work of Modarres-Sadeghi et al. [35], they investigated the post-buckling dynamics of a pinned–pinned pipe conveying steady fluid, with two coupled nonlinear equations for transverse and axial motions being considered.
In order to compare the present result with that of Modarres-Sadeghi et al. [35], a
Frequency analysis
In this section, a frequency analysis will be conducted for the coupled two-pipe system. After neglecting nonlinear terms and damping, one can obtain the linearized forms of the coupled nonlinear governing equations, as follows:
It can be seen from Eqs.
Two pipes conveying steady fluid
In this subsection, the nonlinear dynamics of the two pipes conveying steady fluid are investigated. The complete nonlinear governing equations of the two-pipe system, i.e. Eqs. (30)–(33) are solved, with the dimensionless parameters and initial values defined in Eqs. (41), (42). In this case, the flow velocity of pipe 1 varies in the range of [0, 10], while the flow velocity of pipe 2 is fixed. The flow velocities of pipe 1 and pipe 2 are steady and are denoted as and respectively.
Fig. 8
Conclusions
In this paper, three-dimensional dynamics and dynamical synchronization behaviors of two fluid-conveying pipes coupled with linear springs are investigated. When deriving the governing equations of the coupled pipe system, the geometric nonlinearity due to centerline elongation of the two pipes and extension of the linear springs are considered. A four-mode truncation of the Galerkin’s method is employed to discrete the nonlinear governing equations. A fourth-order Runge–Kutta integration is
CRediT authorship contribution statement
T.L. Jiang: Investigation, Writing – original draft. L.B. Zhang: Writing – review & editing. Z.L. Guo: Writing – review & editing. H. Yan: Validation, Writing – review & editing. H.L. Dai: Supervision, Writing – review & editing. L. Wang: Supervision, Conceptualization, Writing – review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (Nos. 12072119, 12102139 and 11972167).
References (35)
- et al.
Synchronization domains in two coupled neural networks
Commun Nonlinear Sci Numer Simul
(2019) - et al.
Burst mechanisms and burst synchronization in a system of coupled type-i and type-II neurons
Commun Nonlinear Sci Numer Simul
(2020) - et al.
Stochastic transitions between in-phase and anti-phase synchronization in coupled map-based neural oscillators
Commun Nonlinear Sci Numer Simul
(2021) - et al.
A novel synchronization technique for wireless power transfer systems
Electronics
(2018) - et al.
Synchronization of coupled self-excited elastic beams
J Sound Vib
(2009) - et al.
The synchronization of chaotic systems
Phys Rep
(2002) - et al.
Chaos synchronization between linearly coupled chaotic systems
Chaos Solitons Fractals
(2002) - et al.
Connection graph stability method for synchronized coupled chaotic systems
Physica D
(2004) - et al.
Master–slave synchronization of continuously and intermittently coupled sampled-data chaotic oscillators
Commun Nonlinear Sci Numer Simul
(2010) - et al.
Statistical analysis of symbolic dynamics in weakly coupled chaotic oscillators
Commun Nonlinear Sci Numer Simul
(2018)
On the stability of heat exchanger tube bundles, part I: Modified theoretical model
J Sound Vib
On the stability of heat exchanger tube bundles,. part II: Numerical results and comparison with experiments
J Sound Vib
Nonlinear dynamics and synchronization of two coupled pipes conveying pulsating fluid
Acta Mech Solida Sin
Dynamics of cantilevered pipes conveying fluid, part 1: Nonlinear equations of three-dimensional motion
J Fluid Struct
Nonlinear dynamics of extensible fluid-conveying pipes, supported at both ends
J Fluid Struct
Synchronization of chemical micro-oscillators
J Phys Chem Lett
Spatially organized dynamical states in chemical oscillator networks: Synchronization, dynamical differentiation, and chimera patterns
PLoS One
Cited by (4)
A novel retaining clip for vibration reduction of fluid-conveying pipes by piecewise constraints
2024, Mechanical Systems and Signal ProcessingSemi-analytical dynamic modeling of parallel pipeline considering soft nonlinearity of clamp: A simulation and experimental study
2023, Mechanical Systems and Signal ProcessingFatigue life analysis of a slightly curved hydraulic pipe based on Pairs theory
2023, Nonlinear DynamicsAdjacent mode resonance of a hydraulic pipe system consisting of parallel pipes coupled at middle points
2023, Applied Mathematics and Mechanics (English Edition)